The
peaceful coexistence between quantum mechanics and special relativity is
maintained due to the unobservability of hidden variables, and hence theintrinsic unobservability of quantum
non-locality in real time. I prove that if time-reversing quantum measurement is
possible, this coexistence breaks down. Either Lorentz invariance or time-symmetry
must be visibly violated, the more radical option being the inexistence of hidden
variables.

The
EPR argument [[1]], based
on the combined premises of realism, determinism and SR, was in essence an
argument for hidden variables (HVs). Then, Bell’s theorem [[2]] proved that if such HVs exist, they must be
nonlocal. Yet even after Bell-inequality violations were demonstrated, the
violation of SR remained indirect, as no observable superluminal signal can be
sent through the EPR measurements. The proverbial “peaceful coexistence” between
QM and SR [[3]] in due
to this very non-signaling.

Subsequent
theoretical advances related to this issue were relatively minor. Bohm and
Hiley [[4]], seeking to preserve Determinism even under
Bell’s proof, imposed another restriction on HVs: They must exert their
nonlocal effects in some preferred reference frame. This non-relativistic
feature eventually became (see below) central to Bohmian mechanics.

On
the other hand, Elitzur and Dolev [[5]] derived from SR that hidden variables must remain
hidden forever, otherwise their indirect violation of SR would become direct.
This restriction, reminiscent of the “cosmic censorship” conjecture [[6]], equally
prohibits the Bohmian preferred reference frame to be ever detected.

Apart
from these restrictions on HVs, of which Elitzur and Dolev’s [5] renders Bohm’s [4] untestable and leaves Bell’s
[2] as the only one of
experimental significance, no other restrictions were made.

In
this article, I derive a few new restrictions, leading to a new proof of
incompatibility between QM and some basic principles of classical physics.

We
begin with the Proof’s first classical premise (ýi), namely, determinism. Does it still hold beneath
quantum randomness? This debate centers on the notion of “collapse.” When a
particle undergoes, say, a spin measurement, its superposition gives its way to
either an “up” or “down” definite state:

(1),

rendering
the process fundamentally irreversible: While the particle begins as a wave
function split into two halves that move along the two arms of the
Stern-Gerlach magnet (Fig. 1a), the interaction with the detectors leaves a
single particle on one arm, with no trace of the wave function anywhere else (1b).
It is this wave function’s disappearance, by the collapse interpretations, that
makes it impossible to undo the measurement [[7]].

Hidden-variables
interpretations, in contrast, preserve the undetected parts of the wave
function even after measurement. In the many-worlds interpretation [[8]], the above
two possible outcomes are claimed to be two universes splitting from the
original one after the measurement. In Bohmian mechanics [4], these are two halves of
the guide wave, of which one contains the particle while the other is empty but
existing nonetheless. Rather than (1), then, measurement is viewed within
hidden-variables theories as

(2),

the
+ sign between the splitting arrows indicating that both outcomes occur. The single
result actually observed, then, is either the newly-branched universe within
which the newly-branched observer resides, or the wave-function-half within
which the localized particle happened to remain.

By
thus preserving the wave-function’s otherwise mute part, the hidden-variables models
enable in principle a closer look into the question of measurement’s undoing.

Is
it possible, then, to be “unmeasure” a particle, i.e., to time-reverse
its measurement such that the particle resumes its initial superposition? While
a definite answer is beyond this paper’s scope, the consequences of both “yes”
and “no” are obvious:

If quantum measurement cannot be undone in
principle, this would be a fundamental irreversibility, negating
assumptions (ýi) and/or (ýiii) of the Incompatibility Proof and indicating a
microscopic origin of other time-asymmetries [[9]].

If, on the other hand, unmeasurement is
possible, time-symmetry violation return below with vengeance, the other
option being Lorentz-invariance violation.

The
issue, also, is not too remote from present-day technology. A procedure known
as “quantum erasure” [[10]] purports to undo quantum measurement: A
particle undergoing interference is entangled with another superposed particle
while passing through the two routes. Measurement of the latter particle can
reveal which path was taken by the former. Consequently, the original
particle's interference is destroyed. However, by subjecting the other particle
to an appropriate measurement analogous to interference, each particle gives a
"key" that enables observing the other's interference. Upon a closer
inspection, however, no real undoing of measurement is attained with this
method, because no real measurement was made in the first place. Rather,
entanglement occurred, followed (but not undone) by a more subtle measurement.

Elitzur
and Dolev [[11]] proposed a different method of unmeasurement
based on interaction-free measurement. They showed, for the first time, that
just like measurement, unmeasurement too exerts a non-local effect in the
EPR experiment: Unmeasuring one measured particle forces the distant
particle to re-assume its pre-measurement state. Another advance of their
technique is that it is not subject to the no-recording restriction (see
section ý4 below). Katz et al. [[12]] presented a realization of this method,
although not employing it on the EPR setting.

However,
from the HVs viewpoint, one might argue that both the partial measurement and
its undoing involve irreversible absorption of two opposite portions of the “empty”
wave function, hence there was no undoing in the full, time-reversal
sense.

Other
methods, still far from technical feasibility, have been studied [[13]] [[14]]. Like
earlier gedankenexperiments, their ability to yield novel insights gives
an incentive to experimenters to make them realizable.

Annoyingly,
even at the gedanken level, perfect unmeasurement carries yet another price of
a “Faustian” nature: Unmeasurement requires total elimination of any record
of the outcome, including the observer's memory. The smallest trace of a
measurement left unreversed would ruin the entire procedure. This, indeed, is
the case with [10], where the measurement’s
outcome remains unknown.

This
restriction threatens to make the whole issue non-scientific. If “unperformed
measurements have no results” as Peres [[15]] dryly pointed out, then so much the more for
measurements that were performed and then unperformed! Is there any
point in considering unmeasurement if the outcome to be undone must never be
known?

Fortunately,
the EPR setting enables replying to the above question with a resounding “Yes.”
Simply, let a pair of measurements is carried out on two entangled
particles. Here, there is no need to know the outcomes that we seek to undo [[16]]. Suffice
it that we can see, following the unmeasurement of two previously measured
particles, whether they are entangled again, regaining their initial Bell-state.

Fig.
3 depicts such an EPR experiment where the two particles are measured along the
same spin axis, and then undergo unmeasurement. Our question now takes a
simple form: Would the particles resume their entangled state?

Consider,
then, the following standard EPR experiment. An atom with spin 0 ejects two
spin-½ particles, A and B, that fly far apart towards two measuring devices.
These are Stern-Gerlach magnets, randomly aligned along one out of three
coplanar spatial directions α, β and γ, perpendicular to the particle’s flight.
The angles between these directions are chosen so as to maximize violation of
Bell’s inequality: α=0º, β=60º, γ=120º. Each SG magnet measures whether the
particle’s spin is ½ (“up,” ↑) or –½ (“down,” ↓) along the direction chosen.
The probability for correlation between the spin results of each pair (A↑,B↓
or A↓,B↑) depends on the relative angle θ between the measured
spins: p=. This corresponds to a correlation of .

We
envisage, furthermore, an idealized, small and perfectly isolated lab, within
which the above EPR pair undergoes two measurements, without anyone from
outside inspecting the outcomes. Next, the entire process is “undone” within the
lab. Finally, the two particles come out of the lab. If the unmeasurements were
indeed successful, the two particles should be as entangled as they were upon
entering the lab.

With
the above setting we shall first study the restrictions that HVs must obey under
an ordinary pair of measurements. Next we shall consider the case where these
measurements are followed by a pair of two unmeasurements.

The
first restriction may look trivial, but will prove essential for the next
steps. Consider a simple EPR experiment in which both particles undergo spin measurements
in the same direction α. Suppose further that particle A is measured long
before B, its spin turning out to be α↑. B’s spin, by
conservation, must be α↓.

In
order to comply with observational data, HV↑ must be present in 50% of
all measurements and HV↓ in the other 50%. This randomness of the HV↑/↓
distribution, so go HVs theories, is not fundamental but rather apparent, like the
statistical distribution of classical dice throws. Each such a variable is
causally determined by the previous event, as well as causally determining the
consequent spin. Ergo,

Restriction
1:Hidden variables must, by assumption, be deterministic.

6.2.Hidden Variables must be
Nonlocal

It
is to this implicit determinism that Bell amended the next, celebrated
restriction: The EPR correlations cannot be pre-established within the
particles themselves, hence they must be determined by the measurement events.
In terms of Restriction ý6.1, the presence of our presumed
factor
HV↑/↓in one measurement affects not only the nearby particle
but both of them, however mutually remote. Hence,

Restriction
2:Hidden variables must be nonlocal.

6.3.Hidden Variables must Exert their
Nonlocal Effects in some Privileged Reference Frame

Once
nonlocal effects were made inescapable by Bell, all realistic accounts had to assign
them a paradox-free evolution. This is the motivation for Bohm and Hiley’s [4] restriction on the reference
frame within which HVs exert their effects.

Notice,
first, that the uncertainty relations underlying the eigenstates

(3)

inject
a subtle time-asymmetry: Measuring the spin along one direction makes uncertain
the outcome of the subsequent measurement along other directions. This
feature gains further emphasis in the HVs models. In Bohmian mechanics, the
particle “chooses” one Stern-Gerlach arm at the moment the guide wave is split.
Subjecting the outgoing particle to another measurement amounts to loosing the guide
wave’s empty half. Similarly in Everett’s model, consecutive splits of the
universe occur within one another, (Fig. 2) such that within any branch,
part of the wave function is lost. Both models, therefore, introduce a
preferred time-direction to quantum mechanics.

Fig. 2. Branching of the universe during
an EPR experiment according to the Many-Worlds interpretation (from [[17]]).

This
time-asymmetry equally holds whether the two consecutive measurements are carried
out on the same particle or on two distant EPR particles. The correlation
between two spin measurements depends only on the relative angle between them.
Therefore, by Bell’s inequality, the spin value yielded by one particle’s
measurement depends on the choice of spin direction used in the other
particle’s measurement. The bearing of this effect on hidden-variables theories,
imposed by Bohm and Hiley[4], is straightforward: For
any EPR pair of spin measurements,the individual “up” and “down” results
depend on the time-order of the measurement.

True,
Lorentz invariance and quantum indeterminism conspire to camouflage this
temporal feature in the EPR case: It must be impossible to tell which
measurement occurred first. Still, the temporal feature is inescapable for any
realistic account, even in all the realistic “collapse” models (see below). To
quote Bohm and Hiley [4]: “Briefly,
what this means is that there is always a unique frame in which the nonlocal
connections operate instantaneously (p. 285).” Albert [[18], pp.
155-160] is more straightforward: If we measure, say, the α-spin of both EPR particles,
then the individual outcomes depend on which particle is measured first.
The same conclusion is rigorously derived by Barrett [[19], p.
141]: “So the temporal order of the measurements determines the outcome”
(italics original). See also Rae [[20], p. 304],
Callender [[21] and
references therein] and Craig [[22]].

But,
if the two EPR measurements are spacelike separated, what do “first” and
“second” mean? Here hidden variables proponents, perhaps reluctantly, had to go
further: Bohm and Hiley [4] invoked, contra SR, a privileged
reference frame. Popper [[23]] was
quick to point out this move’s implications:

It is
only now, in the light of the new experiments stemming from Bell's work, that
the suggestion of replacing Einstein's interpretation by Lorentz's can be made.
If there is action at a distance, then there is something like absolute space.
If we now have theoretical reasons from quantum theory for introducing absolute
simultaneity, then we would have to go back to Lorentz's interpretation (p. 30).

Since the
exact outcome of the experiment depends on which […] measurement is made first,
the notion of “first” and “second” has an ineliminable physical role in Bohm’s
theory. In the non-relativistic theory, which measurement comes first and which
second is determined by absolute simultaneity. And if one is to transfer the
Bohmian dynamic to a spacetime with a Lorentzian structure, one needs there to
be something fit to play the same dynamical role. Since no such structure is
determined by the Lorentzian metric, the simplest thing to do is to add the
required structure: to add a foliation relative to which the relevant sense of
“first” and “second” is (or “before” and “after”) is defined. The foliation
would then be invoked in the statement of the fundamental dynamical law
governing the particle (p. 162).

Let
us formulate this reasoning in terms of our hypothetical HV↑. By
probability, this variable is assumed to be present in 50% of the measurements.
Hence it must be present in both EPR measurements in 25% of the cases. However,
by angular momentum conservation, the two spins must be opposite. Therefore, to
be paradox-free, the causal priority must go to one measurement, the second being
able only to amplify the previously-determined spin.

Which
measurement, then, is “first” and which “second” when they are
timelike-separated? Fig. 4 shows how all alternatives to non-relativistic
simultaneity either clash with experiment or lead to paradox (see also [[25]]). The nonlocal influence cannot go with any
finite superluminal speed, lest spin-conservation violations ensue. Neither can
the nonlocal influence remain “instantaneous” under Lorentz-transformation. Note
that setting 4b has already been tested by Gisin et al. [[26]] and no Lorentz transformations were observed
on the nonlocal correlations (see also [[27]]). The
debate on “Sutherland’s paradox” [[28], [29]], as
well as the Maudlin-Kastner controversy
[[30]]-[[31]], show that for any realistic temporal account
that allows EPR particles to affect each other’s past, back-and-forth effects
will give rise to causal loops. Bohm’s [4] privileged reference frame,
then, remains the lesser evil.

Admittedly,
the question “what would the results be if the measurements’ time-order was
reversed” is a counterfactual, a method often considered dubious in QM. Recall Peres’
[15] “unperformed measurements
have no results” remark. While Peres considered the imperative “Thou shalt not
think” as a possible remedy, he acknowledged that the EPR argument is based upon
this very kind of counterfactuals. Obviously, it was worth thinking!

While the experimental validation of Bell’s
proof entailed only an indirect violation of SR, the Bohmian preferred frame
brings it closer to a manifest violation. This seems to add rigor to Elitzur
and Dolev’s [5] next
restriction, namely that HVs are hidden not merely for some technical reason.
Rather, relativity dictates that they forever remain what their name
implies.

Consider again the hypothetical HV↑,
whose presence in the measurement causes the particle to assume the spin α↑.
By Restriction 2 (non-locality), HV↑ may reside either in the measuring
apparatus (say, its being positioned on an even or odd number of nanometers
from the source) or in the interaction between the apparatus and the particle
(say, its being performed at an even or odd number of nanoseconds since the
particle’s emission from the source).

Now, as, by Restriction 1, HV↑ is
deterministic, the implicit violation of relativity becomes straightforward: By
merely detectingHV↑’s presence or absence, the decision whether
or not to perform a measurement enables enforcing the particle to assume a
certain spin, thereby sending a superluminal signal to the other measurement. Valentini’s
[[32]] and
Elitzur [[33]]
reached a similar conclusion.[1]

Hence, by SR + Restriction 2,

Restriction 4: Hidden variables
must be hidden forever.

Next, however, I show that, even without
being detectable, hidden variables may lead to violations of SR.

It
is now time to employ the tool of unmeasurement introduced in Section ý3. If we cannot detect HVs,
let us only time-reverse the measurements in which they are involved.

Let,
then, EPR particles A and B undergo spacelike separated measurements, M1
and M2, of their α spins, with a small time interval (in the
absolute frame of Restriction 3) between them. Recall that in 25% of the cases,
the presumed spin-determining variable HV↑ is present in both
M1 and M2. As M1 is
slightly “earlier” than M2, then, by Restriction 4, only M1
can affect A’s spin to be α↑, thereby forcing B to be α↓.
Consequently, M2 can only amplify the state α↓
imposed by M1, despite the presence of HV↑
in M2 too (Fig. 5a).

Next,
perform two unmeasurements M¯1 and M¯2 on the two particles. By
Restriction 1, we assume that HV↑, present in both measuring
devices, takes part also in the two unmeasurements. As M1 has
occurred before M2, now M¯2 must occur
before M¯1. This, by
definition, is true for every deterministic time-reversal. This is also what is
required when the two measurements are performed on the same particle.

We
have, then, two α-spin measurements followed by two unmeasurements, restoring the
original Bell-state:

(4)

So
far so good (Fig. 5a). Things do not go well,
however, when the experiment’s frame is moving (5b). The experimental
setup (e.g., rulers and clocks) undergoes Lorentz transformations, but, by
Restriction 3, the nonlocal effects must not.

The
clash is imminent. By Restriction 3 on HVs, the nonlocal effects
of the measurements and unmeasurements are exerted along the privileged time-planes
t1, t2, t3, t4.
When the frame moves, the simultaneity planes t'1, t'2,
t'3, t'4 of the moving frame become Lorentz-inclined,
deviating from the privileged planes. Consequently, the time-interval between
the spacelike-separated M1 and M2 merely
dilates, but the M¯2- M¯1
interval is reversed. The symmetric sequence M1-M2- M¯2- M¯1 now changes to
the asymmetric M1-M2-M¯1-M¯2, which will ruin the overall time-reversal of
the EPR experiment.

It
should again be pointed out that the meticulous experiment performed by Gisin
and co-workers [26] has tested quantum
nonlocality under relativistic motions of the two particles involved. The
results accord with quantum theory. The present gedankenexperiment adds the
yet-unfeasible challenge of unmeasurement, but now the clash with relativity becomes
inescapable:

The
above anomaly arose from the reasonable assumption that the time-order of the
two unmeasurements must be opposite to that of the earlier measurements. What
if we drop this assumption? A complementary anomaly is imminent.

Consider,
then, the alternative possibility: The individual results of each pair of
measurements depend, by Restriction 3, on the measurements’ time-order, yet the
time order of unmeasurements makes no difference. Whatever spins are
measured andwhatever their results,
unmeasurement always leads back to .

This
assumption is clearly odd, because it is certainly incorrect with respect to a
single particle: If one particle undergoes, say, an α and then a β-spin
measurements

(5),

it
is impossible to unmeasure α before β, as α↑ is no longer valid! It is
therefore hard to believe that the situation will be different with a pair of
EPR measurements.

Even
worse: If, in order to preserve Lorentz invariance, we do not assign importance
to the time-order of the unmeasurements, a time-asymmetry ensues:

The
time-order of the measurements matters in determining the individual spin values
(restriction 3), but the time-order of the unmeasurements does not matter in
bringing them back to their initial state.

Hence:

While
the measurements of some spin directions are non-commuting, their respective unmeasurements always commute.

Let
us inspect this asymmetry in detail. Consider first the time-symmetric evolution
in section ý6.5:

(6)

This
sequence remains invariant under T-reversal: Just read the temporal sequence
backwards. Under this operation, every measurement turns into an unmeasurement
and vice versa, the above sequence remaining unaffected.

Now
assume that the time-order of the unmeasurements is not important, such
that, even under the asymmetric sequence M1-M2-M¯1-M¯2, the same final superposition is obtained:

Under
T-reversal, it is the time-order of the measurements which is
reversed:

(8)

,

and
yet the two measurements’ results and are identical to those of ý(7). Restriction 3, therefore,
holds only in the forward time-direction. Worse, noncommuting measurements turn
out to be commuting!

Admittedly,
the Elitzur-Dolev method [11] as well as the experimental
realization by Katz et al. [12], are immune to the order of
unmeasurements (the order of measurements, of course, being equally devoid of
experimental significance). However, as pointed out above, these methods involve
an irreversible element and may therefore not count as true unmeasurements by
hidden variables theories.

Still,
compared to the manifest Lorentz invariance violation posed by Restriction 5, a
measurement-unmeasurement asymmetry might be the lesser evil:

7.Aiming Higher: Perhaps Question Determinism?

Is
the choice between the non-Lorentz-invariant Scylla and the time-asymmetric Charybdis
forced on us? Not necessarily, for we may go further back along the Incompatibility
Proof and dismiss one of its even more fundamental assumptions.

First,
one may question the most fundamental assumption of classical physics, namely,
realism. In other words, I could add (iv) “realism” as a fourth postulate
in the above Incompatibility Proof, and then mention its dismissal it as a
fourth option at the Proof’s end. I confess that I didn't find this option
worth considering. Even within the Copenhagen school, where quantum phenomena
are denied objective existence, the relations between them must comply
with all physical principles. Bohr’s [[34]] hasty
reply to the EPR argument makes it clear that even if quantum phenomena are
observer dependent, they must never violate special relativity.

What,
then, about the next dreaded price, namely, relinquishing determinism? Perhaps,
so goes this option, the above derivations of Lorentz-non-invariance or time-asymmetry
violation merely indicate that hidden variables do not exist?

Personally,
I always found this option the most tenable one, as well as its broader
implications. Recall Restriction ý6.4, namely, “Hidden variables
must be hidden forever.” This restriction renders HVs a more ghostly entity
than the pre-relativistic ether. The latter was considered merely hard to
observe but there was no theoretical prohibition on its detection.[2] HVs, in
contrast, are entities which SR forbids to be detected or observed. This
places them in a position closer to religion than to science (see Exodus
33, 20).

If,
then, HVs follow the ether to oblivion – if every measurement’s outcome is genuinely
random – then unmeasurement is impossible in the first place: One cannot undo a
measurement if there is no spin-determining variable, hidden or manifest, which
can take part in the undoing. If this is the case, then it is this basic
irreversibility involved in any quantum interaction that constitutes the long
sought-for macroscopic origin of time’s arrow.

Indeed,
even authors working within the collapse framework, such as McCall [17], Leggett [[35]] and
Gisin et al. [26] conclude that collapse
entails a non-Lorentz-invariant physics, hidden forever beneath Restriction 4, in
addition to the penalty of abandoning time-symmetry.

Finally,
let us consider Cramer’s transactional interpretation [[36]] and
Aharonov’s two-vector formalism [25], which describe the EPR
correlations as a spacetime zigzag between the two measurements via the source
in the past. How would these models describe unmeasurement? The
literature related to this model has not dealt with this question yet, but one
thing seems to be clear: Like measurement, the effect of unmeasurement
should also go backwards in time, all the way down till the last measurement.
The conceptual price of this account, however, is high: Different histories
override one another along the same time-sequence every time an unmeasurement
is carried out. Therefore, while these models may preserve the entire
sacred trio of determinism, Lorentz invariance and time-symmetry, this will
demand a radical revision of the very notion of time.

8.Summary

HVs
and time-symmetry are two of QM’s most profound unresolved issues. This paper discusses
gedankenexperiments which combine measurements with unmeasurements. The results
pose two new restrictions on HVs, namely, that they must manifestly violate
either Lorentz invariance or time-symmetry. This unappetizing choice may be
escaped by abandoning HVs (hence determinism) altogether, embracing instead genuine
randomness as a fundamental aspect of physical reality. This would indicate
that the future, just as intuition keeps telling us, is indeed open,
fundamentally nonexistent, subject to some kind of Becoming [[37]] alien
to Relativity. Either way, something very essential seems to be missing in
modern physics’ picture of time.

[1]
Valentini [32]
suggested that quantum nonlocality is hidden only because, in the remote past,
the universe relaxed into statistical equilibrium at the HVs level. To this he
added the even bolder prediction that once a way is found to isolate some
exotic particles that escaped the primordial heat death, superluminal signaling
would become possible, as well as entropy decrease. Elitzur [33]
reached the diametrically-opposed conclusion: The three prohibitions – i)
the relativistic prohibition on superluminal velocities, ii) the quantum
mechanical prohibition on certainty and iii) the thermodynamic
prohibition on entropy decrease – preserve one another in a still-unfathomed
way. Both works, however, agree on one point: If HVs cease to be hidden, SR
will be disproved.